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Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

Yijie Lin

Abstract

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.

Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

Abstract

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.

Paper Structure

This paper contains 16 sections, 59 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Semistable pairs on projective Deligne-Mumford stacks
  3. Preliminaries on projective Deligne-Mumford stacks
  4. The modified Hilbert polynomial
  5. Semistable pairs
  6. Families of coherent sheaves and pairs
  7. Boundedness of the family of semistable pairs
  8. Construction of moduli spaces of semistable pairs
  9. The parameter space, group actions and linearizations
  10. GIT-stability and $\delta$-stability
  11. Moduli spaces of $\delta$-(semi)stable pairs
  12. Variation of moduli spaces
  13. Deformation-obstruction theories and virtual fundamental classes
  14. Deformation and obstruction theories
  15. Virtual fundamental classes
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is a projective scheme $M^{ss}:=M^{ss}_{\mathcal{X}/k}(\mathcal{F}_{0},P,\delta)$ which is a moduli space for the moduli functor $\mathcal{M}^{ss}_{\mathcal{X}/k}(\mathcal{F}_{0},P,\delta)$. Moreover, there is an open subscheme $M^s:=M^s_{\mathcal{X}/k}(\mathcal{F}_{0},P,\delta)$ of $M^{ss}$ w

Theorems & Definitions (135)

  • Theorem 1.1: see Theorem \ref{['GIT-quo1']} and Theorem \ref{['main-result']}
  • Theorem 1.2: see Theorem \ref{['Chamber']}
  • Theorem 1.3: see Theorem \ref{['def-ob1']}
  • Theorem 1.4: see Theorem \ref{['vir-exi-2']}
  • Theorem 1.5: see Theorem \ref{['def-ob2']}
  • Theorem 1.6: see Theorem \ref{['perf-ob1']}, Corollary \ref{['vir-exi-3']} and Theorem \ref{['perf-ob2']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 125 more